Sets of Cardinality 6 Are Not Sum-dominant

TL;DR

This paper proves without computers that sum-dominant sets must have at least 7 elements and identifies the smallest sum-dominant prime set with maximum element 73, improving previous bounds.

Contribution

It provides a human-understandable proof for the minimal size of sum-dominant sets and refines the known bounds for the smallest prime sum-dominant set.

Findings

Sum-dominant sets have at least 7 elements.

The smallest prime sum-dominant set has maximum element 73.

Previous bounds on prime sum-dominant sets are improved.

Abstract

Given a finite set $A\subseteq \mathbb{N}$, define the sum set $$A+A = \{a_i+a_j\mid a_i,a_j\in A\}$$ and the difference set $$A-A = \{a_i-a_j\mid a_i,a_j\in A\}.$$ The set $A$ is said to be sum-dominant if $|A+A|>|A-A|$. Hegarty used a nontrivial algorithm to find that $8$ is the smallest cardinality of a sum-dominant set. Since then, Nathanson has asked for a human-understandable proof of the result. However, due to the complexity of the interactions among numbers, it is still questionable whether such a proof can be written down in full without computers' help. In this paper, we present a computer-free proof that a sum-dominant set must have at least $7$ elements. We also answer the question raised by the author of the current paper et al about the smallest sum-dominant set of primes, in terms of its largest element. Using computers, we find that the smallest sum-dominant set of primes has $73$ as its maximum, smaller than the value found before.